Cones and frustum slides

2. What is a Cone?
 A solid or hollow object that tapers from a
circular or roughly circular base to a point.

3. Types of Cones
 Right Cone - A cone that has its apex aligned
directly above the center of its base.
 Oblique Cone – A cone that has its apex not
aligned above the center of its base
Right Cone Oblique Cone
http://www.mathope
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4. Volume of Cone
Formula:
Volume = 𝜋𝑟2
ℎ
r = radius of circular base
h = height of the cone
Can be used for both right and oblique
cone.

5. Total Surface Area
Formula:
𝑇𝑆𝐴 = 𝜋rs + 𝜋𝑟2
= 𝜋𝑟(𝑠 + 𝑟)
r = radius of circular base
s = slant height of the cone
This formula CANNOT be used for oblique
cone. (There are NO formula to find TSA of
oblique cone)

6. Example 1
Find out the Volume and the
Total Surface Area of the
cone.
< Given: 𝜋 =
22
7
>
Volume =
1
3
𝜋𝑟2
ℎ
=
1
3
×
22
7
× 72
×10
= 513.333𝑐𝑚3

7. s = 102 + 72
= 149
Total Surface Area
= 𝜋𝑟 𝑠 + 𝑟
=
22
7
× 7( 149 + 7)
= 22 149 + 7
= 422.544𝑐𝑚2

8. What is a Conical Frustum?
A conical frustum is a frustum created by
slicing the top off a cone (with the cut
made parallel to the base).

9. Volume of Conical Frustum
METHOD 1
 Formula:
V =
𝜋ℎ
3
(𝑅2
+ 𝑅𝑟 + 𝑟2
)
h = height of the frustum
r = radius of the circular top of the frustum
R = radius of the circular base of the frustum

10. Total Surface Area of Conical
Frustum
𝑠 = (𝑅 − 𝑟)2 + ℎ2
 Formula:
Lateral Surface Area:
TSA = 𝜋𝑠 (𝑅 + 𝑟)
Circular Surface Area:
(Top) CSA = 𝜋𝑟2
(Bottom) CSA = 𝜋𝑅2
Total Surface Area:
TSA = 𝜋𝑠 𝑅 + 𝑟 + 𝜋𝑟2 + 𝜋𝑅2
= 𝜋 (𝑅 − 𝑟)2 + ℎ2 𝑅 + 𝑟 + 𝜋𝑟2 + 𝜋𝑅2

11. Example 2
Find out the Volume and the
Total Surface Area of the
frustum.
< Given: 𝜋 =
22
7
>
Volume =
𝜋ℎ
3
(𝑅2 + 𝑅𝑟 + 𝑟2)
=
22
7
×14
3
(72
+ 35 +
52)
=
44
3
× 109
= 1598.667𝑐𝑚3

12. S = (7 − 5)2+142
= 10 2
Total Surface Area
= 𝜋𝑠 𝑅 + 𝑟 + 𝜋𝑅2
+ 𝜋𝑟2
=
22
7
× 10 2 7 + 5 +
22
7
×
72
+
22
7
× 52
= 533.361 + 154 + 78.571
= 765.932𝑐𝑚2

13. Example 3
Diagram shows cone A and
frustum B. Ratio of slant height
cone A to slant height frustum
B is 1:2 . Given that radius of
cone A is 4cm and its volume
is 16π𝑐𝑚3
. Find out the
volume and the total surface
area of frustum B in terms of
π.

14. Height of cone A:
Volume =
1
3
𝜋𝑟2ℎ
16𝜋𝑐𝑚3 =
1
3
× 𝜋 × 42 × ℎ
16𝑐𝑚3 =
16
3
ℎ
ℎ = 16 ×
3
16
ℎ = 3𝑐𝑚
Slant height of cone:
ℓ = 𝑟2 + ℎ2
= 42 + 32
= 25
= 5𝑐𝑚

15. Ratio of slant height cone A to slant height frustum B

16. Solution 1:
Volume of frustum B
=
1
3
𝜋𝑅2 𝐻 −
1
3
𝜋𝑟2ℎ
=
1
3
𝜋 × 122 × 9 −
1
3
𝜋 × 42 × 3
= 432𝜋 − 16𝜋
= 416𝜋𝑐𝑚3 Solution 2:
Volume of frustum B
=
𝜋 𝐻 − ℎ
3
(𝑅2
+ 𝑅𝑟 + 𝑟2
)
=
𝜋 9 − 3
3
122 + 48 + 42
= 2𝜋 × 208
= 416𝜋𝑐𝑚3

17. Total Surface Area of frustum B
= 𝜋 (𝑅 − 𝑟)2 + (𝐻 − ℎ)2 𝑅 + 𝑟 + 𝜋𝑟2 +
𝜋𝑅2
= 𝜋 12 − 4 2 + 9 − 3 2 12 + 4 +
𝜋(4)2 + 𝜋(12)2
= 𝜋10 16 + 16𝜋 + 144𝜋
= 160𝜋 + 160𝜋
= 320𝜋𝑐𝑚2